Lecture 7. Binary Search Trees and Red-Black Trees

Transcription

1 Lecture Binary Search Trees and Red-Black Trees

2 Announcements HW released! (Due Friday)

3 Today: binary search trees Brief foray into data structures! See CS 166 for more! What are binary search trees? You may remember these from CS 106B Why are they good? Why are they bad? this will lead us to Self-Balancing Binary Search Trees Red-Black trees.

4 Why are we studying self-balancing BSTs? 1. The punchline is important: A data structure with O(log(n)) INSERT/DELETE/SEARCH 2. The idea behind Red-Black Trees is clever It s good to be exposed to clever ideas. Also it s just aesthetically pleasing.

5 Motivation for binary search trees We ve been assuming that we have access to some basic data structures: (Sorted) linked lists HEAD (Sorted) arrays

6 Sorted linked lists O(1) insert/delete (assuming we have a pointer to the location of the insert/delete): HEAD O(n) search/select: 6 HEAD

7 Sorted Arrays O(n) insert/delete: O(log(n)) search, O(1) select: Search: Binary search to see if is in A. Select: Third smallest is A[].

8 The best of both worlds Sorted Arrays Linked Lists Binary Search Trees* Search O(log(n)) O(n) O(log(n)) Insert/Delete O(n) O(1) O(log(n))

9 Tree terminology This node is the root 5 This is a node. It has a key (). Today all keys are distinct. Both children of are NIL 1 The left child of is 2 The right child of is These nodes are leaves.

10 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node. Example of building a binary search tree:

11 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node. Example of building a binary search tree:

12 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node. Example of building a binary search tree:

13 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node. Example of building a binary search tree:

14 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node. Example of building a binary search tree:

15 Aside: this should look familiar kinda like QuickSort

16 Binary Search Trees It s a binary tree so that: Every LEFT descendent of a node has key less than that node. Every RIGHT descendent of a node has key larger than that node Binary Search Tree 1 NOT a Binary Search Tree

17 SEARCH in a Binary Search Tree definition by example EXAMPLE: Search for 4. EXAMPLE: Search for 4.5 It turns out it will be convenient to return 4 in this case (that is, return the last node before we went off the tree) 1 Write pseudocode (or actual code) to implement this! Ollie the over-achieving ostrich

18 INSERT in a Binary Search Tree x = EXAMPLE: Insert 4.5 INSERT(key): x = SEARCH(key) if key > x.key: Make a new node with the correct key, and put it as the right child of x. if key < x.key: Make a new node with the correct key, and put it as the left child of x. if x.key == key: return

19 DELETE in a Binary Search Tree EXAMPLE: Delete 2 5 DELETE(key): x = SEARCH(key) if x.key == key:.delete x x = 2 This is a bit more complicated

20 DELETE in a Binary Search Tree several cases (by example) say we want to delete Case 1: if is a leaf, just delete it. 2 This triangle is a cartoon for a subtree 2 We won t write down pseudocode for this try to do it yourself! Case 2: if has just one child, move that up. Ollie the over-achieving ostrich

21 DELETE in a Binary Search Tree ctd. Case : if has two children, replace with it s immediate successor. (aka, next biggest thing after ) How do we find the immediate successor? SEARCH(.right, ) How do we remove it when we find it? Run DELETE for one of the previous two cases..1 Wait, what if it s THIS case? (Case ). It s not.

22 How long do these operations take? SEARCH is the big one. Everything else just calls SEARCH and then does some small O(1)-time operation. 5 Time = O(depth of tree) Trees have depth O(log(n)). Done! Lucky the lackadaisical lemur.

23 Wait This is a valid binary search tree. The version with n nodes has depth n, noto(log(n)) Could such a tree show up? In what order would I have to insert the nodes? Inserting in the order 2,,4,5,6,,8 would do it. So this could happen.

24 What to do? How often is every so often in the worst case? It s actually pretty often! Keep track of how deep the tree is getting. Ollie the over-achieving ostrich If it gets too tall, re-do everything from scratch. At least Ω(n) every so often. Other ideas?

25 Idea 1: Rotations No matter what lives underneath A,B,C, this takes time O(1). (Why?) Maintain Binary Search Tree (BST) property, while moving stuff around. YOINK! X Y Y Y C A B X A X A B C B C CLAIM: this still has BST property.

26 This seems helpful YOINK!

27 Does this work? Whenever something seems unbalanced, do rotations until it s okay again. Even for me this is pretty vague. What do we mean by seems unbalanced? What s okay? Lucky the Lackadaisical Lemur

28 Idea 2: have some proxy for balance Maintaining perfect balance is too hard. Instead, come up with some proxy for balance: If the tree satisfies [SOME PROPERTY], then it s pretty balanced. We can maintain [SOME PROPERTY] using rotations. There are actually several ways to do this, but today we ll see

29 Red-Black Trees A Binary Search Tree that balances itself! No more time-consuming by-hand balancing! Be the envy of your friends and neighbors with the time-saving 5 Maintain balance by stipulating that black nodes are balanced, and that there aren t too many red nodes It s just good sense!

30 Red-Black Trees these rules are the proxy for balance Every node is colored red or black. The root node is a black node. NIL children count as black nodes. Children of a red node are black nodes. For all nodes x: all paths from x to NONE s have the same number of black nodes on them

31 Examples(?) Yes! Every node is colored red or black. The root node is a black node. NONE children count as black nodes. Children of a red node are black nodes. For all nodes x: all paths from x to NONE s have the same number of black nodes on them. No! No! No!

32 Why??????? This is pretty balanced. The black nodes are balanced The red nodes are spread out so they don t mess things up too much. We can maintain this property as we insert/delete nodes, by using rotations

33 Let s build some intuition! This is pretty balanced Lucky the lackadaisical lemur To see why, intuitively, let s try to build a Red-Black Tree that s unbalanced. One path could be twice as long as all the others if we pad it with red nodes. Conjecture: the height of a red-black tree is at most 2 log(n)

34 That turns out the be basically right. [proof sketch] Say there are b(x) black nodes in any path from x to NONE. (including x). Claim: Then there are at least 2 b(x) 1 nodes in the subtree underneath x. [Proof by induction on board if time] Then: n 2 $ %&&' 1 2 +,-.+/ Rearranging: using the Claim b(root) >= height/2 because of RBTree rules. n ' 1 height 2log (n + 1) y x

35 Okay, so it s balanced but can we maintain it? Yes! For the rest of lecture: sketch of how we d do this. See CLRS for more details.

36 Inserting into a Red-Black Tree 6 Make a new red node. Insert it as you would normally. What if it looks like this? 6 Example: insert 0 0

37 Inserting into a Red-Black Tree 6 Make a new red node. Insert it as you would normally. Fix things up if needed. 6 Example: insert 0 What if it looks like this? No! 0

38 Inserting into a Red-Black Tree 6 Make a new red node. Insert it as you would normally. Fix things up if needed. 6 Example: insert 0 What if it looks like this? Can t we just insert 0 as a black node? 0 No!

39 Inserting into a Red-Black Tree -1 Make a new red node. Insert it as you would normally. Fix things up if needed. 6 What if it looks like this? Example: insert Instead recolor like this. Need to argue: RB-Tree properties still hold. What about the red root? if 6 is actually the root, color it black. Else, recursively re-color up the tree. Now the problem looks like this, where I m inserting ish

40 Inserting into a Red-Black Tree 6 Make a new red node. Insert it as you would normally. Fix things up if needed. 0 6 Example: Insert 0. What if it looks like this? Actually, this can t happen? It might happen that we just turned 0 red from the previous step. Or it could happen if is actually NIL.

41 Recall Rotations Maintain Binary Search Tree (BST) property, while moving stuff around. YOINK! X Y Y Y C A B X A X A B C B C CLAIM: this still has BST property.

42 Inserting into a Red-Black Tree Make a new red node. Insert it as you would normally. Fix things up if needed. 6 What if it looks like this? YOINK! 6 Need to argue that if RB-Tree property held before, it still does

43 That s basically it A few things still left to check for INSERT! Anything else that might happen looks basically like what we just did. Formally dealing with the recursion. You check these! (or see CLRS) DELETEis similar. The punchline: Plucky the pedantic penguin Red-Black Trees always have height at most 2log(n+1). As with general Binary Search Trees, all operations are O(height) So all operations are O(log(n)).

44 Conclusion: The best of both worlds Sorted Arrays Linked Lists Balanced Binary Search Trees Search O(log(n)) O(n) O(log(n)) Insert/Delete O(n) O(1) O(log(n))

45 Recap Balanced binary trees are the best of both worlds! But we need to keep them balanced. Red-Black Trees do that for us. We get O(log(n))-time INSERT/DELETE/SEARCH Clever idea: have a proxy for balance Next time Hashing!